We now consider a time-dependent large-scale flow, by allowing for lateral oscillations of the flow pattern
adopted in the previous section, which is enough to produce chaotic Lagrangian trajectories of fluid particles [1]}, [2]}. More explicitly, the flow field will now be specified by Eq. (REF ), i.e. Eq. (REF ) with \(\Psi _{s} = 0\) . The amplitude and pulsation of the roll oscillation are respectively set to \(s=L_z/5\)  and \(\omega =\pi U/L_z\) , corresponding to a fraction of the roll size and a period comparable to the advective timescale \(L_z/U\) , a choice that has been shown to be optimal to enhance chaotic diffusion [3]}, [1]}, [5]}, [6]}. Note that we do not allow for vertical oscillations, in order to keep the top and bottom boundaries of our domain at fixed vertical positions. Figure REF  shows two snapshots of the population field at different times in the constant per-capita growth-rate regime. These visualizations suggest that the dynamics are fairly similar to those in the stationary-flow case, although horizontal symmetry is now broken due to the lateral oscillations of the flow.
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